The thinning of the liquid layer over a probe in two-phase flow

Abstract

The draining of the thin water film that is formed between a two dimensional, infinite, initially flat oil-water interface and a smooth, symmetric probe, as the interface is advected by a steady and uniform flow parallel to the probe axis, is modelled using classical fluid dynamics.

The governing equations are nondimensionalised using values appropriate to the oil extraction industry. The bulk flow is driven by inertia and, in some extremes, surface tension while the viscous effects are initially confined to thin boundary layers on the probe and the interface. The flow in the thin water film is dominated by surface tension, and passes through a series of asymptotic regimes in which inertial forces are gradually overtaken by viscous forces. For each of these regimes, and for those concerning the earlier stages of approach, possible solution strategies are discussed and relevant literature reviewed.

Consideration is given to the drainage mechanism around a probe which protrudes a fixed specified distance into the oil. A lubrication analysis of the thin water film may be matched into a capillary-static solution for the outer geometry using a slender transition region if, and only if, the pressure gradient in the film is negative as it meets the static meniscus. The remarkable result is that, in practice, there is a race between rupture in the transition region and rupture at the tip. The analysis is applicable to the case of a very slow far field flow and offers significant insight into the non-static case.

Finally, a similar approach is applied to study the motion of the thin water film in the fully inviscid approximation, with surface tension and a density contrast between the fluids.